As the swim coach at Gardner Bullis, Gabriela selects which athletes will participate in the state-wide swim relay. The relay team swims $\frac{2}{3}$ of a mile in total, with each team member responsible for swimming $\frac{2}{9}$ of a mile. The team must complete the swim in $\frac{3}{4}$ of an hour. How many swimmers does Gabriela need on the relay team?
Solution: To find out how many swimmers Gabriela needs on the team, divide the total distance ( $\frac{2}{3}$ of a mile) by the distance each team member will swim ( $\frac{2}{9}$ of a mile). $ \dfrac{{\dfrac{2}{3} \text{ mile}}} {{\dfrac{2}{9} \text{ mile per swimmer}}} = {\text{ number of swimmers}} $ Dividing by a fraction is the same as multiplying by the reciprocal. The reciprocal of ${\dfrac{2}{9} \text{ mile per swimmer}}$ is ${\dfrac{9}{2} \text{ swimmers per mile}}$ $ {\dfrac{2}{3}\text{ mile}} \times {\dfrac{9}{2} \text{ swimmers per mile}} = {\text{ number of swimmers}} $ $ \dfrac{{2} \cdot {9}} {{3} \cdot {2}} = {\text{ number of swimmers}} $ Reduce terms with common factors by dividing the $2$ in the numerator and the $2$ in the denominator by $2$ $ \dfrac{{\cancel{2}^{1}} \cdot {9}} {{3} \cdot {\cancel{2}^{1}}} = {\text{ number of swimmers}} $ Reduce terms with common factors by dividing the $9$ in the numerator and the $3$ in the denominator by $3$ $ \dfrac{{1} \cdot {\cancel{9}^{3}}} {{\cancel{3}^{1}} \cdot {1}} = {\text{ number of swimmers}} $ Simplify: $ \dfrac{{1} \cdot {3}} {{1} \cdot {1}} = {3} $ Gabriela needs 3 swimmers on her team.